2002

AbstractType-I intermittencies are common phenomena that are often observed in the neighborhood of periodic windows when a control parameter is varied. These intermittencies usually have a single reinjection channel, that is, a single type of laminar phase was observed. Recently, type-I intermittencies with two reinjection channels were reported in several systems. In this paper, it will be shown that type-I intermittencies with n channels of reinjection are associated with the coexistence of n stable periodic orbits that are mapped into each other under a symmetry. A procedure to build type-I intermittency with n reinjection channels using the n-fold cover of an image system is presented. Cases up to n=3 are explicitly given with the covers of the centered Rössler system.

AbstractThe fact that oscillations can be induced in studies of the maintenance of the electrical potential of frog skin by addition of lithium allowed evaluation of several parameters fundamental to the functioning of the system in vivo (e.g. relative volumes of internal compartments, characteristic times of ionic exchanges between compartments). A realistic model was thus proposed under the form of a set of ordinary differential equations. In the past, numerical simulations using such a model reproduced the periodic experimental oscillations and was able to provide an explanation for the global synchronised oscillations of the whole skin. In that paper, new numerical simulations reproduce the non-periodic oscillations which were observed two decades ago, but not reproduced by the model. Moreover, the dynamical process under which all the local oscillators are synchronised is explained in terms of a tangent bifurcation.

AbstractWhen a dynamical system is investigated from a time series, one of the most challenging problems is to obtain a model that reproduces the underlying dynamics. Many papers have been devoted to this problem but very few have considered the influence of symmetries in the original system and the choice of the observable. Indeed, it is well known that there are usually some variables that provide a better representation of the underlying dynamics and, consequently, a global model can be
obtained with less difficulties starting from such variables. This is connected to the problem of observing the dynamical system from a single time series. The roots of the nonequivalence between the dynamical variables will be investigated in a more systematic way using previously defined observability indices. It turns out that there are two important ingredients which are the complexity of the coupling between the dynamical variables and the symmetry properties of the original system. As will be mentioned, symmetries and the choice of observables also has important consequences in other problems such as synchronization of nonlinear oscillators.

AbstractIt is shown that nonlinear global models identified from a single time series can be used to reproduce the same sequence of bifurcations of the original system. This has been observed for simulated and real data and for both difference equation and differential equation models, thus suggesting some generality. The results reported in this paper are of a practical character and seem to have some bearing not only on the important subject of estimating bifurcation diagrams from data, but also in model validation problems since some models can reproduce the bifurcation sequence of a system even when such models do not settle to the original attractor at first. In this case, models which would be dismissed are shown to display consistent dynamic information about the original system, as illustrated by a simulated and a real data example. An additional example that uses real data is provided in which the original bifurcation sequence is recovered by the addition of multiplicative noise with increasing variance.

Abstract: A three-variable biochemical prototype involving two enzymes with autocatalytic regulation proposed by Decroly and Goldbeter (1987) is analyzed using a topological approach. A two-branched manifold, a so-called template, is thus identified. For certain control parameter values, this template is a horseshoe template with a global torsion of two half-turns. This implies that the bifurcation diagram can be described using the usual sequences associated with a unimodal map with a differentiable maximum as well exemplified by the logistic map. Moreover, a type-I intermittency associated with a saddle-node bifurcation is exhibited. The dynamics is also investigated from a single time series to determine whether it is possible to investigate the dynamics of this biochemical model from the measure of a single concentration.

Abstract: A fairly realistic three species food chain model based on the Leslie-Gower scheme is investigated by using tools borrowed from the nonlinear dynamical systems theory. It is observed that two co-existing attractors may be generated by this ecological model. A type-I intermittency is characterized and a homoclinic orbit is found.

Abstract: A fairly realistic three species food chain model based on Lotka-Volterra and Leslie-Gower schemes is investigated assuming that just a single scalar time series available. The paper uses tools borrowed from the theory of nonlinear dynamical systems. The quality of the different phase portraits reconstructed is tested. Such a situation would arise in practice whenever only a single species is counted. It is found that the dynamical analysis can be safely performed when a single species involved in the food chain is counted if many thousands of observations are available. If not, a global model can be obtained from the available data and subsequently used to produce all the data required for a detailed analysis. In this case, however, the choice of which species to consider in order to obtain a model is crucially important.